Question

use the relation \\(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\\) to determine the limit. \\(\lim_{\theta \to 0} \frac{4 \sin \sqrt{2} \theta}{\sqrt{2} \theta}\\) select the correct answer below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. \\(\lim_{\theta \to 0} \frac{\dots \sin \sqrt{2} \theta}{\dots} = \square\\) (type an integer or a simplified fraction.) \\(\bigcirc\\) b. the limit does not exist.

Explanation:

Step1: Let \( u = \sqrt{2}\theta \)

As \( \theta \to 0 \), \( u \to 0 \) (since \( \sqrt{2}\) is a constant, multiplying by \( \theta \) which approaches 0 makes \( u \) approach 0).

Step2: Rewrite the limit

The original limit is \( \lim_{\theta \to 0} \frac{4\sin\sqrt{2}\theta}{\sqrt{2}\theta} \). We can factor out the constant 4: \( 4\lim_{\theta \to 0} \frac{\sin\sqrt{2}\theta}{\sqrt{2}\theta} \). Now substitute \( u = \sqrt{2}\theta \), so the limit becomes \( 4\lim_{u \to 0} \frac{\sin u}{u} \).

Step3: Apply the given limit relation

We know that \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \) (from the given relation \( \lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1 \), here \( u \) is just a variable, so the limit holds). So we substitute this value into our expression: \( 4\times1 \).

Answer:

A. \( \lim_{\theta \to 0} \frac{4\sin\sqrt{2}\theta}{\sqrt{2}\theta} = \boxed{4} \)